CN110866354A - Optimized design method of polymer vascular stent structure considering scale effect - Google Patents
Optimized design method of polymer vascular stent structure considering scale effect Download PDFInfo
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Abstract
The invention relates to a scale effect-considered polymer intravascular stent structure optimization design method, belongs to the field of interventional medical instruments, and relates to a scale effect-considered polymer intravascular stent structure optimization design method. The method comprises the steps of obtaining initial sample points by defining an optimization problem and adopting an optimized Latin hypercube method on the basis of considering the influence of a scale effect on the mechanical behavior of a support, and establishing an approximate function relation between a target function and a design variable by using a Kriging agent model in combination with a finite element method. And based on the approximate function and the criterion of improving the expected point adding, combining the genetic algorithm and the sequence quadratic programming algorithm to carry out parallel operation to obtain the improved design of the bracket structure. And outputting the final optimization design result of the support when the optimal solution of the objective function meets the convergence condition. The invention makes up the defect of neglecting the scale effect in the prior art, introduces the novel constitutive relation into the finite element calculation, adopts the optimization algorithm to improve the calculation precision and the calculation efficiency, and has strong applicability.
Description
Technical Field
The invention belongs to the technical field of interventional medical instruments, and particularly relates to a scale effect considered optimal design method of a polymer vascular stent structure.
Background
At present, cardiovascular and cerebrovascular diseases are the leading killers of death, and the blood flow blockage caused by vascular stenosis is the main reason for triggering the diseases. The polymer intravascular stent is used as an interventional medical appliance implanted into a human body, can effectively avoid angiostenosis, reduce morbidity, and meet a plurality of performance indexes such as radial support rigidity, radial elastic recovery, axial shortening, surface coverage rate and the like simultaneously in order to ensure the safety and reliability of the stent in a long-term service process and meet the structural design requirements. Meanwhile, with the continuous development of polymer fine manufacturing technology, more and more researches show that the polymer material has scale effect under the micro-scale. Unlike conventional continuous medium mechanics, when a material is deformed non-uniformly on a micro-scale, its apparent elastic modulus or hardness is greatly increased. The microporous structure is formed inside the polymer intravascular stent due to material degradation in the service process, and the effect exists when the structure is bent and twisted. On the premise of considering the scale effect, how to establish an effective method and realize the multi-objective optimization design of the polymer vascular stent structure becomes a problem to be solved at present.
The invention patent application number of terminal end et al is 201711292177.3, and discloses a method, a device and electronic equipment for simulating the expansion of a virtual stent in a blood vessel, which realizes the real-time monitoring of the expansion condition of the stent in the blood vessel, but the invention only displays the three-dimensional expansion process of the stent, does not consider the influence of the scale effect on the mechanical property of the stent, and does not make a corresponding structure optimization design method. The invention discloses a manual intervention type aortic valve stent with radial support force change, which is disclosed as 201910181781.1 in the invention patent application number of Yanyogo et al.
The invention patent application number 201910178807.7 of Lihongxia et al discloses a drug sustained release simulation and optimization method for a drug eluting stent, which optimizes parameters of the drug eluting stent by finite element simulation and surrogate model optimization methods, and can shorten the research and development period of the drug eluting stent, but the method does not consider the influence of scale effect and stent structure on the mechanical performance of the stent, and lacks applicability in the aspect of stent structure optimization design.
Disclosure of Invention
Aiming at the defects of the prior art, the influence of the scale effect on the mechanical property of the degradable vascular stent is not considered, the structural design of the vascular stent mostly takes experience adjustment as the main part, and the vascular stent designed in the prior art lacks general applicability in the face of complex implantation environment, so that the invention provides the optimal design method of the polymer vascular stent structure considering the scale effect. The influence of the scale effect and the support structure on the mechanical performance of the support is considered, the calculation accuracy and the calculation efficiency are improved by using the optimization algorithm, and the applicability is strong.
The method adopts the technical scheme that the method is an optimized design method of the polymer vascular stent structure considering the scale effect, obtains initial sample points by defining an optimization problem and adopting an optimized Latin hypercube method on the basis of considering the influence of the scale effect on the stent mechanical behavior, and establishes an approximate functional relation between a target function and a design variable by using a Kriging agent model in combination with a finite element method. And based on the approximate function and the criterion of improving the expected point adding, combining the genetic algorithm and the sequence quadratic programming algorithm to carry out parallel operation to obtain the improved design of the bracket structure. And outputting the final optimization design result of the support when the optimal solution of the objective function meets the convergence condition. The method comprises the following specific steps:
step one, introducing even stress m and curvature strain χ based on a Cosserat continuous medium theory, and establishing a polymer material constitutive relation considering a scale effect:
wherein the Cauchy stress sigma, the even stress m, the Cauchy strain epsilon and the curvature strain chi are vector forms:
σ＝[σxxσyyσzzτxyτyxτyzτzyτzxτxz]T(2)
m＝[mxxmyymzzmxymyxmyzmzymzxmxz]T(3)
ε＝[εxxεyyεzzεxyεyxεyzεzyεzxεxz]T(4)
χ＝[χxxχyyχzzχxyχyxχyzχzyχzxχxz]T(5)
the generalized elastic stiffness matrix D is:
wherein D isuuAnd DωωThe rigidity matrix D is respectively related to the displacement and rotation of any material point in the polymer blood vessel stent material1、D2、D3Is defined as:
wherein, Λ is E v/(1 + v) (1-2 v) and μ are Lame constants, E is elastic modulus, v is Poisson's ratio, μcIs the second shear modulus,/tIs the characteristic length, l, associated with the torsion of the materialbIs a characteristic length associated with material bending;
generalized stress sigmagAnd generalized strain εgThe equivalent of (a) is:
by constructing a polymer material somatic cell model with micro-cavities, the equivalent constitutive relation function of the polymer material is obtained as follows
Wherein u isiAnd FiIs the displacement and force of the upper boundary of the somatic cell, S is the upper boundary area of the somatic cell, heIs the length of the somatic cell in the i direction, VeIs the somatic volume.
Step two, defining the structure optimization design problem of the polymer degradable vascular stent, wherein the structure optimization design problem comprises design variables, design targets and constraint conditions;
the radial elastic retraction, the axial shortening and the surface coverage rate in the expansion period are minimized under the conditions that the stent has enough supporting force in the supporting period and has enough supporting force retention rate in the blood vessel remodeling period.
Optimally designing the polymer intravascular stent by taking the length a and the width b of the diamond holes as design variables; the mathematical expression of the structure optimization design problem is as follows:
wherein, ER, AS and SC are respectively the radial elastic retraction rate, the axial retraction rate and the surface coverage rate of the vascular stent, u is the structural design variable of the polymer vascular stent,
andurespectively are the upper limit and the lower limit of the design variable, I is the number of the design variable,
is the minimum radial support force, μ, required for a given support period0Is the minimum holding force required for a given period of vascular remodelingAnd (4) rate.
Extracting 16 design variable initial sample points in a design space by adopting an optimized Latin hypercube method;
1) dividing each dimensional interval of the design variable into 16 intervals with equal probability;
2) randomly extracting a point in each interval of each dimension;
3) randomly extracting one point from the points extracted in the step 2) for each dimension, and forming a vector by using the points;
analyzing the sample points extracted in the third step by adopting a three-dimensional finite element model, wherein the finite element model comprises four parts, namely a blood vessel, a thrombus plaque, a polymer blood vessel stent and a balloon, and because the whole model has symmetry, 1/12 of the whole model, namely a circumferential 1/6 and an axial 1/2 are selected for simulation calculation in order to improve the calculation speed; performing grid division on the model by using ANSYS17.0, wherein the blood vessel, the thrombus plaque and the polymer vascular stent adopt 8-node Solid 185 entity units, and the balloon adopts 4-node Shell181 Shell units;
the load and boundary conditions applied to the finite element model are as follows:
1) applying symmetry constraints on the symmetry planes of the polymer vascular stent, the blood vessel and the thrombus plaque, and simultaneously constraining circumferential rotation and axial movement of the balloon;
2) the load is added by applying an internal pressure changing along with time in the saccule, the internal pressure change curve is divided into three parts of linear loading, constant loading and linear unloading, and the blood vessel support changes along with the change of the internal pressure of the saccule.
Solving the target function response value of each initial sample point, and selecting a group of samples corresponding to the minimum response value as an optimization starting point;
based on sample information, an approximate functional relation between a target function and a design variable is obtained by using a Kriging agent model, and the agent model comprises a regression part and a nonparametric part
Where β is the regression coefficient, f (x) is the regression polynomial determined by training the samples, and the global approximation is simulated in the design space, and z (x) is the random distribution error, which provides an approximation simulation of the local bias, with the statistical properties:
E[z(x)]＝0 (14)
wherein x isiAnd xjIs any two sample points, R (θ, x)i,xj) Theta is a parameter for representing the spatial correlation of each sample point, and the correlation function R is expressed by a continuous differentiable Gaussian function
In the formula, nvIs the number of known design variables and,
and
respectively, known sample point xiAnd xjThe kth component of (a);
from a known set of sample points
And a set of responses
For any newly added sample point xnewThe response value can be predicted from a linear combination of the response set Y of known sample points:
the estimation error is:
wherein F ═ F1,f2,…fn]T,Z＝[z1,z2,…zn]T. At the same time, the average of the prediction errors must be equal to zero, i.e. the average of the prediction errors must be equal to zero, since the unbiased requirement needs to be met
Then there are:
FTc-f＝0 (21)
the estimated variance is:
after finishing, the method comprises the following steps:
wherein the content of the first and second substances,
this equation characterizes the new sample point xnewCorrelation in space with other known sample points. Then, a coefficient c is obtained by minimizing the estimation variance of the estimation value, and the optimization model is as follows:
and (3) solving by using a Lagrange multiplier method:
c＝R-1[r-F(FTR-1F)-1(FTR-1r-f)](26)
the estimated variance of the estimated value is obtained as follows:
get the new sample point xnewEstimated value of (a):
using regression problems
The generalized least squares estimate of (a) can be obtained:
β*＝(FTR-1F)-1FTR-1Y (29)
margin expression R gamma considering the regression problem*＝Y-Fβ*It is possible to obtain:
the matrix F, R and vector Y are derived from a set S of known sample points, for a new sample point xnewIf f (x) can be obtainednew) And r (x)new) Then, a new sample point x can be obtainednewThe estimated response value of (2), only the unknown parameters need to be solved at this time
And the parameter θ in R, the likelihood function of y (x) being, due to obeying a normal distribution:
taking the logarithm of the above equation and removing the constant term yields:
relate to
The standing value of (c) can be obtained:
further obtaining:
the maximum likelihood estimate of theta is obtained by solving the maximum of the log-likelihood function, i.e.
And finally, obtaining an approximate functional relation between the target function and the design variable.
Step six, solving the expected EI improvement corresponding to the objective function F (x)
E[I(x)]＝σ(x)[uΦ(u)+φ(u)](36)
Wherein the content of the first and second substances,
and σ2(x) Is the mean and mean square error, Y, corresponding to any design point x in spaceminIs the current optimum value, phi is the regularization probability distribution function, phi is the probability density function;
based on the approximate function and the improved expected EI, the genetic algorithm and the sequence quadratic programming algorithm are utilized to carry out the combined operation to obtain the improved design result of the bracket,
1) based on EI value, obtaining optimal solution x by adopting genetic algorithm and sequence quadratic programming algorithmk1 *And xk2 *；
2) Based on approximate function, obtaining optimal solution x by adopting genetic algorithm and sequence quadratic programming algorithmk3 *And xk4 *。
Step seven, respectively solving x by using Kriging agent modelk1 *、xk2 *、xk3 *And xk4 *Predicted value of (2)
And
separately computing x by finite element methodk1 *、xk2 *、xk3 *And xk4 *Is the objective function value F (x)k1 *)、F(xk2 *)、F(xk3 *) And F (x)k4 *) And selecting the better one as the current optimal solution F (x)k *)。
Step eight, checking convergence conditions as follows:
1)
2)
3)|F(xk *)-F(xk-1 *)|≤ε2(40)
wherein, Deltar、ε1And ε2Is given convergence accuracy, YmaxAnd YminIs the maximum and minimum response value in the sample points, and k is the number of iteration steps of the optimization procedure.
1) When the objective function is optimal, the solution F (x)k *) When the convergence condition is not satisfied, adding an improved design point xk1 *、xk2 *、xk3 *And xk4 *Continuing to iteratively solve in the sample until the sample is satisfiedA convergence condition;
2) when the objective function is optimal, the solution F (x)k *) Outputting the optimized design result x of the polymer vascular stent structure when the convergence condition is metk *。
The invention has the advantages that the design method establishes a novel constitutive relation considering the polymer material dimension effect, and introduces the novel constitutive relation into finite element calculation to form a novel calculation method for detecting the mechanical property of the intravascular stent. Aiming at the optimization design of the high nonlinear and multidimensional large-scale design space of the balloon-expandable intravascular stent, the approximate function relation between the objective function and the design variable is constructed by using a Kriging surrogate model, and the calculation efficiency is obviously improved. The genetic algorithm and the sequence quadratic programming algorithm are combined, the advantages of high local optimization efficiency and strong global optimization capability of the genetic algorithm and the sequence quadratic programming algorithm are fully exerted, and the optimization process is enabled to efficiently approach the global optimal solution. The invention makes up the defect of neglecting the scale effect in the prior art, improves the calculation precision and the calculation efficiency by adopting an optimization algorithm, and has strong applicability.
Drawings
FIG. 1 is a flow chart of a method for optimally designing a polymer vascular stent structure by considering a scale effect.
Fig. 2 shows the process of the degradation rate and the support performance of the polymer blood vessel stent.
Fig. 3 is a geometry of a polymeric vascular stent. Wherein, a is the prismatic length of the polymer vascular stent, b is the prismatic width of the polymer vascular stent, w is the rib width of the polymer vascular stent, and d is the thickness of the polymer vascular stent.
FIG. 4 is a finite element model of a polymer vascular stent in-service process.
Fig. 5 is a time-dependent change in balloon pressure.
Fig. 6 is a cloud of radial displacement profiles of the optimized and original stents.
Detailed Description
The following detailed description of the embodiments of the invention is provided in connection with the accompanying drawings and the accompanying claims.
FIG. 1 is a flow chart of a method for optimally designing a polymer vascular stent structure by considering a scale effect. The design method comprises the following specific steps:
step one, based on a Cosserat continuous medium theory, introducing even stress m and curvature strain chi, and establishing a polymer material constitutive relation considering a scale effect through a formula (1), wherein Cauchy stress sigma, even stress m, Cauchy strain epsilon and curvature strain chi are vector forms and are respectively represented by formulas (2) to (5), a generalized elastic stiffness matrix D is represented by a formula (6), and a stiffness matrix D is specifically represented by formulas (7) to (9)1、D2、D3To express, and then to obtain generalized stress σgAnd generalized strain εgThe equivalent of (c):
then passing through a generalized elastic rigidity matrix D and a rigidity matrix D1、D2、D3Equations (9) - (12) establish the elastic constitutive relation equation (11) of the polymer material considering the scale effect.
And step two, defining the structure optimization design problem of the polymer degradable vascular stent, wherein the structure optimization design problem comprises design variables, design targets and constraint conditions.
The polymer vascular stent shown in figure 3 is optimally designed by taking the length a and the width b of the diamond holes as design variables. It can be described specifically as: the radial elastic retraction, the axial shortening and the surface coverage rate in the expansion period are minimized under the conditions that the stent has enough supporting force in the supporting period and has enough supporting force retention rate in the blood vessel remodeling period.
The mathematical expression of the structure optimization design problem is as follows:
extracting 16 design variable initial sample points in a design space by adopting an optimized Latin hypercube method, which specifically comprises the following steps:
1) dividing each dimensional interval of the design variable into 16 intervals with equal probability;
2) randomly extracting a point in each interval of each dimension;
3) and randomly extracting one point from the points extracted in the step 2) for each dimension, and forming the points into a vector.
And step four, analyzing the sample points extracted in the step three by using the three-dimensional finite element model shown in the figure 4, wherein the finite element model comprises four parts, namely a blood vessel, a thrombus plaque, a polymer blood vessel stent and a balloon, and because the whole model has symmetry, 1/12 of the whole model, namely the circumferential 1/6 and the axial 1/2 are selected for simulation calculation in order to improve the calculation speed.
The structural parameters of the model are: the length of the blood vessel is 8.4mm, and the thickness is 0.1 mm; the length of the thrombus plaque is 7mm, and the thickness of the proximal end is 0.3 mm; the length of the polymer intravascular stent is 6.875mm, the thickness d is 0.15mm, and the tendon width w is 0.15 mm; the balloon length was 7.6mm and the thickness was 0.05 mm.
And carrying out meshing on the model by using ANSYS17.0, wherein the blood vessel, the thrombus plaque and the polymer blood vessel stent adopt 8-node Solid 185 Solid units, and the balloon adopts 4-node Shell181 Shell units.
The material properties of the model are shown in table 1.
Material Properties of the model of Table 1
Fig. 2 shows the process of the degradation rate and the support performance of the polymer blood vessel stent.
Step five, simulating the service process of the polymer intravascular stent, and applying loads and boundary conditions to the finite element model:
1) applying symmetry constraints on the symmetry planes of the polymer vascular stent, the blood vessel and the thrombus plaque, and simultaneously constraining circumferential rotation and axial movement of the balloon;
2) the load is added by applying the internal pressure shown in fig. 5 to the balloon interior, and the internal pressure change curve is divided into three portions of linear loading, constant loading P0.9148 MPa, and linear unloading.
The objective function response value of each initial sample point was solved using ANSYS17.0, and the specific results are shown in table 2.
TABLE 2 results of finite element calculations for initial sample points
And step six, selecting a group of samples corresponding to the minimum response value as an optimization starting point. Based on the sample information, an approximate functional relationship between the objective function and the design variable shown in formula (13) is obtained by using a Kriging agent model, and the detailed derivation process is shown in formulas (14) - (35).
Seventhly, solving an improved expected EI value corresponding to the objective function F (x) by using formulas (36) and (37), and performing combined operation by using a genetic algorithm and a sequence quadratic programming algorithm to obtain an improved design result of the stent based on an approximate function and the improved expected EI, wherein the improved design result specifically comprises the following steps of:
1) based on EI value, obtaining optimal solution x by adopting genetic algorithm and sequence quadratic programming algorithmk1 *And xk2 *；
2) Based on approximate function, obtaining optimal solution x by adopting genetic algorithm and sequence quadratic programming algorithmk3 *、xk4 *。
Step eight, respectively solving x by using Kriging agent modelk1 *、xk2 *、xk3 *And xk4 *Predicted value of (2)
And
separately computing x by finite element methodk1 *、xk2 *、xk3 *And xk4 *Is the objective function value F (x)k1 *)、F(xk2 *)、F(xk3 *) And F (x)k4 *) And selecting the better one as the current optimal solution F (x)k *)。
The convergence conditions shown by equations (38) to (40) are examined:
1) when the objective function is optimal, the solution F (x)k *) When the convergence condition is not satisfied, adding an improved design point xk1 *、xk2 *、xk3 *And xk4 *Continuing iterative solution in the sample until a convergence condition is met;
2) when the objective function is optimal, the solution F (x)k *) Outputting the optimized design result x of the polymer vascular stent structure when the convergence condition is metk *。
Finally, the optimal design results are obtained after 21 iterations, as shown in table 3.
TABLE 3 Performance comparison of optimized scaffolds to original and reference scaffolds
Wherein, the reference bracket only reduces the width w and the thickness d of the rib by 0.02mm on the basis of the original bracket, and other parameters are unchanged.
Compared with the original stent, the radial elastic retraction rate of the optimized stent is almost the same as that of the original stent, which shows that the optimized stent has the radial support capability similar to that of the original stent even if the rib width w of the optimized stent is reduced by 13.33% and the thickness d is reduced by 11.76%. The width w and the thickness d of the rib of the optimized stent are smaller, and the surface coverage rate of the rib is 16.05 percent smaller than that of the original stent, so that the neointimal hyperplasia probability is reduced, the risk of restenosis in the stent is reduced, and meanwhile, the flexibility of the optimized stent is better than that of the original designed stent, and the transportation of the stent in a blood vessel is facilitated. For the axial shortening rate of the stent, the axial shortening rate of the three stents is below 20%, and the constraint condition defined by the optimization problem is met.
In general, the optimized bracket has radial supporting capacity similar to that of the originally designed bracket on the premise of obviously reducing the width w and the thickness d of the rib, the surface coverage rate of the bracket is reduced, and the flexibility of the bracket is improved, so that the comprehensive service performance of the bracket is improved.
Furthermore, compared to the reference stent with simply reduced width w and thickness d of the stent rib, the radial elastic retraction rate of the optimized stent is 22.48% less than that of the reference stent although the optimized stent has the same width and thickness, which indicates that the optimized stent has better radial supporting capability; and the surface coverage of the optimized stent is also slightly less than that of the reference stent, indicating that the likelihood of a long term in-stent restenosis risk for the optimized stent is less.
Fig. 6 depicts a cloud of radial displacement profiles of an optimized stent versus a reference stent after balloon unloading. As can be seen, the radial displacement of the optimized stent as a whole is greater than the radial displacement of the reference stent after balloon unloading. This is because, when the balloon is removed, the stent will undergo radial recoil under the action of the elastic deformation of the vessel and the stent itself, while the radial elastic recoil of the optimized stent is smaller than that of the reference stent, so that the radial displacement on the optimized stent is greater at the last moment than that of the reference stent, which also proves that the optimized stent has better expansion performance and supporting capability.
Claims (1)
1. A polymer blood vessel support structure optimization design method considering scale effect, on the basis of considering the influence of scale effect on support mechanical behavior, by defining optimization problem, obtaining initial sample point by adopting optimized Latin hypercube method, combining finite element method, and establishing approximate function relation between objective function and design variable by using Kriging agent model; based on an approximate function and an improved expected point adding criterion, combining a genetic algorithm and a sequence quadratic programming algorithm to carry out parallel operation to obtain an improved design of the support structure; outputting a final optimization design result of the support when the optimal solution of the objective function meets the convergence condition; the method comprises the following specific steps:
step one, introducing even stress m and curvature strain χ based on a Cosserat continuous medium theory, and establishing a polymer material constitutive relation considering a scale effect:
wherein the Cauchy stress sigma, the even stress m, the Cauchy strain epsilon and the curvature strain chi are vector forms:
σ＝[σxxσyyσzzτxyτyxτyzτzyτzxτxz]T(2)
m＝[mxxmyymzzmxymyxmyzmzymzxmxz]T(3)
ε＝[εxxεyyεzzεxyεyxεyzεzyεzxεxz]T(4)
χ＝[χxxχyyχzzχxyχyxχyzχzyχzxχxz]T(5)
the generalized elastic stiffness matrix D is:
wherein D isuuAnd DωωThe rigidity matrix D is respectively related to the displacement and rotation of any material point in the polymer blood vessel stent material1、D2、D3Is defined as:
wherein, Λ is E v/(1 + v) (1-2 v) and μ are Lame constants, E is elastic modulus, v is Poisson's ratio, μcIs the second shear modulus,/tIs the characteristic length, l, associated with the torsion of the materialbIs a characteristic length associated with material bending;
generalized stress sigmagAnd generalized strain εgThe equivalent of (a) is:
by constructing a polymer material somatic cell model with micro-cavities, the equivalent constitutive relation function of the polymer material is obtained as follows:
wherein u isiAnd FiIs the displacement and force of the upper boundary of the somatic cell, S is the upper boundary area of the somatic cell, heIs the length of the somatic cell in the i direction, VeIs the somatic volume;
step two, defining the structure optimization design problem of the polymer degradable vascular stent, wherein the structure optimization design problem comprises design variables, design targets and constraint conditions;
the radial elastic retraction, the axial shortening and the surface coverage rate in the expansion period are minimized under the conditions that the stent has enough supporting force in the supporting period and has enough supporting force retention rate in the blood vessel remodeling period;
optimally designing the polymer intravascular stent by taking the length a and the width b of the diamond holes as design variables; the mathematical expression of the structure optimization design problem is as follows:
wherein ER, AS and SC are vascular stents respectivelyThe radial elastic retraction rate, the axial shortening rate and the surface coverage rate of the polymer vascular stent, u is a design variable of the polymer vascular stent structure,
andurespectively are the upper limit and the lower limit of the design variable, I is the number of the design variable,
is the minimum radial support force, μ, required for a given support period0Is the minimum holding force required for a given period of vascular remodeling;
extracting 16 design variable initial sample points in a design space by adopting an optimized Latin hypercube method;
1) dividing each dimensional interval of the design variable into 16 intervals with equal probability;
2) randomly extracting a point in each interval of each dimension;
3) randomly extracting one point from the points extracted in the step 2) for each dimension, and forming a vector by using the points;
analyzing the sample points extracted in the third step by adopting a three-dimensional finite element model, wherein the finite element model comprises four parts, namely a blood vessel, a thrombus plaque, a polymer blood vessel stent and a balloon, and because the whole model has symmetry, 1/12 of the whole model, namely a circumferential 1/6 and an axial 1/2 are selected for simulation calculation in order to improve the calculation speed; performing grid division on the model by using ANSYS17.0, wherein the blood vessel, the thrombus plaque and the polymer vascular stent adopt 8-node Solid 185 entity units, and the balloon adopts 4-node Shell181 Shell units;
the load and boundary conditions applied to the finite element model are as follows:
1) applying symmetry constraints on the symmetry planes of the polymer vascular stent, the blood vessel and the thrombus plaque, and simultaneously constraining circumferential rotation and axial movement of the balloon;
2) the load is added by applying an internal pressure which changes along with time in the saccule, the internal pressure change curve is divided into three parts of linear loading, constant loading and linear unloading, and the vascular stent changes along with the change of the internal pressure of the saccule;
solving the target function response value of each initial sample point, and selecting a group of samples corresponding to the minimum response value as an optimization starting point;
based on sample information, an approximate functional relation between a target function and a design variable is obtained by using a Kriging agent model, and the agent model comprises a regression part and a nonparametric part
Wherein β is a regression coefficient, f (x) is a regression polynomial determined by training samples, and approximate simulation is performed on the global in the design space, and z (x) is a random distribution error, and provides approximate simulation of local deviation, and the statistical properties are as follows:
E[z(x)]＝0 (14)
wherein x isiAnd xjIs any two sample points, R (θ, x)i,xj) For the correlation function, θ is a parameter for characterizing the spatial correlation of each sample point, and the correlation function R is expressed as a continuous differentiable gaussian function:
in the formula, nvIs the number of known design variables and,
and
are respectively knownSample point xiAnd xjThe kth component of (a);
from a known set of sample points
And a set of responses
For any newly added sample point xnewThe response value can be predicted from a linear combination of the response set Y of known sample points:
the estimation error is:
wherein F ═ F1,f2,…fn]T,Z＝[z1,z2,…zn]T(ii) a At the same time, the average of the prediction errors must be equal to zero, i.e. the average of the prediction errors must be equal to zero, since the unbiased requirement needs to be met
Then there are:
FTc-f＝0 (21)
the estimated variance is:
after finishing, the method comprises the following steps:
wherein the content of the first and second substances,
this equation characterizes the new sample point xnewCorrelation in space with other known sample points; then, a coefficient c is obtained by minimizing the estimation variance of the estimation value, and the optimization model is as follows:
and (3) solving by using a Lagrange multiplier method:
c＝R-1[r-F(FTR-1F)-1(FTR-1r-f)](26)
the estimated variance of the estimated value is obtained as follows:
get the new sample point xnewEstimated value of (a):
using regression problems
The generalized least squares estimate of (a) can be obtained:
β*＝(FTR-1F)-1FTR-1Y (29)
margin expression R gamma considering the regression problem*＝Y-Fβ*Obtaining:
the matrix F, R and vector Y are derived from a set S of known sample points, for a new sample point xnewIf f (x) can be obtainednew) Andr(xnew) Then, a new sample point x can be obtainednewThe estimated response value of (2), only the unknown parameters need to be solved at this time
And the parameter θ in R, the likelihood function of y (x) being, due to obeying a normal distribution:
taking the logarithm of the above equation and removing the constant term yields:
relate to
The standing value of (c) yields:
further obtaining:
obtaining a maximum likelihood estimate of theta by solving the maximum of the log-likelihood function, i.e.
Finally, obtaining an approximate functional relation between the target function and the design variable;
step six, solving the expected EI improvement corresponding to the objective function F (x)
E[I(x)]＝σ(x)[uΦ(u)+φ(u)](36)
Wherein the content of the first and second substances,
and σ2(x) Is the mean and mean square error, Y, corresponding to any design point x in spaceminIs the current optimum value, phi is the regularization probability distribution function, phi is the probability density function;
based on the approximate function and the improved expected EI, the genetic algorithm and the sequence quadratic programming algorithm are utilized to carry out the combined operation to obtain the improved design result of the bracket,
1) based on EI value, obtaining optimal solution x by adopting genetic algorithm and sequence quadratic programming algorithmk1X and xk2*；
2) Based on approximate function, obtaining optimal solution x by adopting genetic algorithm and sequence quadratic programming algorithmk3X and xk4*；
Step seven, respectively solving x by using Kriging agent modelk1*、xk2*、xk3X and xk4Predicted value of
And
separately computing x by finite element methodk1*、xk2*、xk3X and xk4Objective function value of F (x)k1*)、F(xk2*)、F(xk3X) and F (x)k4X) and selecting the better one as the current optimal solution F (x)k*)；
Step eight, checking convergence conditions as follows:
1)
2)
3)|F(xk*)-F(xk-1*)|≤ε2(40)
wherein, Deltar、ε1And ε2Is given convergence accuracy, YmaxAnd YminThe maximum and minimum response values in the sample points, k being the number of iteration steps of the optimization program;
1) when the objective function is optimal, the solution F (x)kX) does not satisfy the convergence condition, add the improved design point xk1*、xk2*、xk3X and xk4Continuing iterative solution in the sample until a convergence condition is met;
2) when the objective function is optimal, the solution F (x)kX) when the convergence condition is satisfied, outputting an optimized design result x of the polymer vascular stent structurek*。
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